Numerical Approximation of a Nonlinear Mixed Type Functional Differential Equation

We begin with a brief review of our previous work with autonomous and non-autonomous linear MTFDEs using collocation, least squares and finite element methods. Then we focus on the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The considered model describes the conduction of neuroelectric signals in a myelinated nerve axon (composed by a membrane and nodes) . In this case, when the membrane is depolarized at a node, the myelin tends not to depolarize the adjacent region of membrane, but instead it appears to jump to the next node to excite the membrane there, as described by the authors of [1]. As a consequence, the variation in time of the electric potential at each node depends on the electric potential of the neighbour nodes and is modeled by a first order nonlinear functional differential equation with deviated arguments. Following the approach introduced previously for linear MTFDEs, we propose and analyse a new computational method for the solution of this problem.

References

1. H. Chi, J.Bell and B. Hassard, Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory, J.Math.Biology, 24 (1986), 583-601.